For Defective Matrices
Generalized eigenvectors are needed to form a complete basis of a defective matrix, which is a matrix in which there are fewer linearly independent eigenvectors than eigenvalues (counting multiplicity). Over an algebraically closed field, the generalized eigenvectors do allow choosing a complete basis, as follows from the Jordan form of a matrix.
In particular, suppose that an eigenvalue λ of a matrix A has an algebraic multiplicity m but fewer corresponding eigenvectors. We form a sequence of m eigenvectors and generalized eigenvectors that are linearly independent and satisfy
for some coefficients, for . It follows that
The vectors can always be chosen, but are not uniquely determined by the above relations. If the geometric multiplicity (dimension of the eigenspace) of λ is p, one can choose the first p vectors to be eigenvectors, but the remaining m − p vectors are only generalized eigenvectors.
Read more about this topic: Generalized Eigenvector
Famous quotes containing the word defective:
“The study and knowledge of the universe would somehow be lame and defective were no practical results to follow.”
—Marcus Tullius Cicero (10643 B.C.)